Derivative Formulas for Simple Functions and Roots

The basic derivative formulas are (c)′ = 0, (xⁿ)′ = n xⁿ⁻¹ for every real n, (e^x)′ = e^x, (ln x)′ = 1/x, (sin x)′ = cos x and (cos x)′ = −sin x; roots are handled by writing them as powers. In brief: (x⁵)′ = 5x⁴, (√x)′ = 1/(2√x), (1/x)′ = −1/x² and (3x⁴ − 2x + 7)′ = 12x³ − 2. The derivative of a constant times a function is the constant times the derivative.

Computing a derivative from the definition is slow. A small table of formulas, combined with a few rules (sum, constant multiple, and later the product, quotient and chain rules), covers all elementary functions. This guide gives the power rule with a proof, the formulas for roots and reciprocals, for exponential and logarithmic functions and for sine and cosine, shows how to differentiate polynomials and expressions with roots, how to find higher derivatives and gives a complete table.

What is the power rule?

(xⁿ)′ = n xⁿ⁻¹ for every real number n (for x where the expressions are defined). Special cases: (x)′ = 1, (x²)′ = 2x, (x³)′ = 3x², and (c)′ = 0 for a constant c.

Proof for a positive integer n: [(x + h)ⁿ − xⁿ]/h = [n xⁿ⁻¹h + C(n, 2)xⁿ⁻²h² + … + hⁿ]/h = n xⁿ⁻¹ + C(n, 2)xⁿ⁻²h + … + hⁿ⁻¹ → n xⁿ⁻¹.

How do you differentiate roots and reciprocals?

Rewrite roots and reciprocals as powers with fractional or negative exponents and apply the power rule.

• √x = x^1/2: (√x)′ = (1/2)x^−1/2 = 1/(2√x).
• ³√x = x^1/3: (x^1/3)′ = (1/3)x^−2/3 = 1/(3 ³√(x²)).
• ⁿ√x: (x^1/n)′ = (1/n)x^{1/n − 1}.
• 1/x = x⁻¹: (1/x)′ = −x⁻² = −1/x².
• 1/xⁿ: (x⁻ⁿ)′ = −n x⁻ⁿ⁻¹ = −n/xⁿ⁺¹.

Example 1

f(x) = 4√x − 6/x² + 5: write f(x) = 4x^1/2 − 6x⁻² + 5. Then f′(x) = 2x^−1/2 + 12x⁻³ = 2/√x + 12/x³. At x = 4: 2/2 + 12/64 = 1.1875.

Example 2

g(x) = x²√x = x^5/2, so g′(x) = (5/2)x^3/2 = (5/2)x√x. And h(x) = (x³ + 1)/x = x² + 1/x, so h′(x) = 2x − 1/x².

Example 3: a cube root

f(x) = ³√x at x = 8: f′(8) = 1/(3 · 8^2/3) = 1/(3 · 4) = 1/12. At x = 0, the derivative is not defined (a vertical tangent), though the function is continuous.

What are the sum and constant multiple rules?

• (c f)′ = c f′.
• (f + g)′ = f′ + g′ and (f − g)′ = f′ − g′.

Together, differentiation is a linear operation. A polynomial is differentiated term by term.

Example 4

(3x⁴ − 2x³ + 5x − 7)′ = 12x³ − 6x² + 5. And (x²/2 + 4x)′ = x + 4. And a polynomial of degree n has a derivative of degree n − 1.

Example 5: slope at a point

f(x) = x³ − 3x² + 2: f′(x) = 3x² − 6x. The slope at x = 2 is 12 − 12 = 0 (a horizontal tangent). At x = 3: 27 − 18 = 9.

What are the derivatives of exponential, logarithmic and trigonometric functions?

• (e^x)′ = e^x, (a^x)′ = a^x ln a.
• (ln x)′ = 1/x, (log_a x)′ = 1/(x ln a).
• (sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = 1/cos²x.
• (arcsin x)′ = 1/√(1 − x²), (arctan x)′ = 1/(1 + x²).

Example 6

(5e^x − 3 ln x)′ = 5e^x − 3/x. And (2^x)′ = 2^x ln 2, so at x = 3 the slope is 8 · 0.6931 = 5.545. And (4 sin x + cos x)′ = 4 cos x − sin x.

What are higher derivatives?

The second derivative f″ is the derivative of f′, the third f‴, and so on. In motion, if s(t) is the position, s′ is the velocity and s″ is the acceleration. The second derivative measures how the slope changes: it is positive where the graph curves upward (concave up) and negative where it curves downward.

Example 7

f(x) = x⁴ − 2x²: f′ = 4x³ − 4x, f″ = 12x² − 4, f‴ = 24x, f⁽⁴⁾ = 24 and f⁽⁵⁾ = 0. A polynomial of degree n has all derivatives of order above n equal to 0.

Example 8: motion

s(t) = t³ − 6t² + 9t (meters). v(t) = 3t² − 12t + 9 = 3(t − 1)(t − 3), and a(t) = 6t − 12. The velocity is zero at t = 1 and t = 3. At t = 2: v = −3 m/s and a = 0 m/s².

Table of derivative formulas

Where do the formulas come from?

Fermat’s method of adequality (1630s) and Newton’s work on fluxions gave the derivative of powers. Newton’s generalised binomial theorem (1665) extended the power rule to fractional exponents. Euler, in the eighteenth century, derived the derivatives of exponential, logarithmic and trigonometric functions from his analytic definitions of these functions. Notation by Leibniz (dy/dx) and Lagrange (f′) is used today.

Where are they used?

Derivatives of polynomials give velocities and accelerations in physics, marginal costs in economics, and slopes in engineering. The exponential derivative (e^kx)′ = k e^kx says that the growth rate is proportional to the quantity, the basis of models of population and radioactive decay. The formulas for roots appear in the speed of a falling object or the rate at which the side of a square changes with its area: if A = s², then s = √A and ds/dA = 1/(2√A).

Which mistakes are common?

Differentiating (xⁿ) as n^x

xⁿ is a power function (the variable is the base): (xⁿ)′ = n xⁿ⁻¹. For the exponential n^x: (n^x)′ = n^x ln n.

Forgetting to rewrite roots

Convert √x to x^1/2 before differentiating. For 1/√x = x^−1/2, the derivative is −(1/2)x^−3/2.

Wrong exponent after differentiation

(x^1/2)′ = (1/2)x^−1/2, with the exponent 1/2 − 1 = −1/2.

Differentiating a product term by term

(x · x)′ is not 1 · 1. For products use the rule in the next article, or simplify first: (x²)′ = 2x.